1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 8.3 Describing and Analyzing Data

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Calculate numerical descriptors of data, such as measures of center, standard deviation, percentiles, and z-score

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Describing and Analyzing Data Displaying data in a clear and informative way is certainly an important and necessary step in research. Just as important is describing the data numerically, so that they can be compared and analyzed.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Arithmetic Mean The mean is the sum of all of the data values divided by the number of data points. Formally, the formula for the population mean is where x i is the i th data value and N is the number of data values in the population.

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Arithmetic Mean (cont.) The formula for the sample mean is where x i is the i th data value and n is the number of data values in the sample.

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Mean A sample of the number of sick days employees at Witt’s Insurance Agency took during last year is listed below. Calculate the mean of the sample data. 14, 5, 7, 11, 9, 7, 12, 6 Solution There are 8 pieces of sample data, so in order to find the sample mean, add all the values together and divide by 8.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Mean (cont.) Therefore, the mean of this sample is 8.9.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Mean – Working Backwards Exam scores 70, 75, 83, 90, and one more to go. What is needed on fifth exam to attain an 80 average?

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Median The median of a data set is the middle value in an ordered array of the data.

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Finding the Median A VO 2 max score is the maximum amount of oxygen that one's body can transport and use during exercise. It is measured in liters of oxygen per minute (L/min). Given the following VO 2 max scores for 12 women, find the median score. 28.3, 27.7, 23.0, 25.5, 27.1, 26.94, 27.0, 27.52, 26.8, 27.2, 26.97, 27.53

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Finding the Median (cont.) Solution First, put the data in ascending numerical order. 23.0, 25.5, 26.8, 26.94, 26.97, 27.0, 27.1, 27.2, 27.52, 27.53, 27.7, 28.3 Since there are 12 pieces of data, the median will be the value between the middle two data points, 27.0 and 27.1. To find this, add the two together and divide by two.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Finding the Median (cont.) Once again, the value of this “average” is not a member of the data set. However, it is a typical value in the sense that it is located in the middle of the data set when it is arranged numerically.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Mode The mode is the value in the data set that occurs most frequently.

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Finding the Mode Find the mode of each of the following sets of data. State if the data set is unimodal, bimodal, multimodal, or has no mode. a.Preferred color of cell phone cases among students lemon, gunmetal, violet, turquoise, lime, violet, lemon, orange, red, lemon, pink, violet, lime, violet, lemon, pink, gunmetal, red, turquoise, violet, violet, gunmetal, turquoise, red, violet, turquoise, orange, pink, violet, violet, turquoise, violet, pink

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Finding the Mode (cont.) b.Favorite football jersey number 32, 18, 99, 12, 7, 10, 28, 56, 13, 16, 19, 51, 23, 78 c.Ages of children at the community playground one afternoon 12, 4, 2, 7, 8, 4, 10, 6, 5, 7, 7, 4, 3 d.Number of ATM withdrawals per hour at the downtown branch of University Bank 10, 13, 9, 13, 9, 14, 10, 14

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Finding the Mode (cont.) Solution a.The color violet occurs more than any other color, so the mode is violet. This data set is unimodal. b.Each value occurs only once, so there is no mode. c.The values 4 and 7 both occur an equal number of times, which is more than any other value. Thus, the set is bimodal with the modes 4 and 7. d.Be careful here. Since each value occurs the same number of times, there is no mode in this data set.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Outlier An outlier is a data value that is extreme compared with the rest of the data values in the set.

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding the Mean, Median, and Mode Given the following data set, find the mean, median, and mode, and decide which measure of center you think best describes the data set. 16, 44, 15, 48, 14, 77, 11, 84, 26, 61, 15 Solution To find the mean, add up all of the data values and divide by 11 (the number of data values).

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding the Mean, Median, and Mode (cont.) To find the median, arrange the values in ascending order and find the middle value. Median = 26 The mode is the most commonly occurring value. Notice that 15 occurs twice, while all other values occur only once. Therefore, the mode is 15.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding the Mean, Median, and Mode (cont.) Although there is a mode, because it only occurs twice while all the other data points occur once, this is not the best descriptor of the “average” piece of data. A mode of 15 does not accurately reflect the middle of the data set since the data ranges from 11 to 84. That leaves the mean and the median. Since there are not any outliers it’s appropriate to use the mean of the data as the measure of center for this data.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Find the mean, median, and mode of the following data. 8, 12, 10, 11, 13, 12, 15, 9, 11, 16 Answer: Mean: 11.7; Median: 11.5; Mode: 11, 12

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Mean, Median, or Mode? Which one to use? Which measurement best characterizes the middle of a data set? 1.If it’s not a measurement, but something like a color or flavor, then use the Mode. (Which answer happened most often?) 2.If there are outliers – some weird very low or very high data values – then Median is better. 3.Otherwise, the Mean is the bets measure of center.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Range The range is the difference between the largest and smallest values in the data set, which tells you the distance covered on the number line between the two extremes. range = maximum data value – minimum data value

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding the Range Find the range of the following sets of data. a.The number of students enrolled as computer science majors over the past 12 semesters 5, 21, 54, 33, 12, 14, 36, 40, 27, 29, 37, 22 b.The number of shoppers at a gas station downtown Monday through Sunday one week 1007, 1010, 1006, 1005, 1054, 1021, 1005

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Finding the Range (cont.) Solution a.The maximum value is 54 and the minimum value is 5, so the range is 54  5 = 49. b.The maximum value for the data set is 1054 and the minimum value is 1005, so the range is also 1054  1005 = 49.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Standard Deviation The standard deviation is a measure of how much we might expect a member of the data set to differ from the mean. The formula for finding the population standard deviation is where x i is the i th data value,  is the population mean, and N is the size of the population.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Standard Deviation (cont.) For a sample, the standard deviation is where x i is the i th data value, is the sample mean, and n is the sample size.

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Standard Deviation by Hand Calculate the sample standard deviation for a sample of nine ages of students working with a university theater production of Macbeth. 17, 21, 18, 18, 24, 19, 21, 20, 28 Solution When calculating the standard deviation by hand, we need to first note the sample size n and find the sample mean With n = 9, the mean is

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Standard Deviation by Hand (cont.) Note that we will round the mean to the nearest hundredth in an effort to minimize any error introduced from rounding. When calculating standard deviation by hand, it’s helpful to use a table like Table 1 and build up to the formula.

30. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Standard Deviation by Hand (cont.) Table 1: Sample Standard Deviation xixi  17  3.67 13.47 210.330.11 18  2.67 7.13 18  2.67 7.13 243.3311.09 19  1.67 2.79 210.330.11 20  0.67 0.45 287.3353.73 

31. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Standard Deviation by Hand (cont.) We are now ready to substitute the values into the formula for the sample standard deviation.

32. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 6: Calculating Standard Deviation by Hand (cont.) So, the sample standard deviation of ages is approximately 3.5. In other words, the age of the average student in the sample is about 3.5 years different (either younger or older) from the mean age of 20.67.

33. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator Use your TI-30XIIS/B or TI-83/84 Plus calculator to find the standard deviation of the following data sets. a.The following data represent the average number of Tweets per day posted on Twitter for a sample of 24 college students. Table 2: Tweets Per Day 0.842.220.62.836.712.1 18.66.35.511.33.70.5 1.23.714.99.47.39.5 16.011.14.75.68.910.2

34. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) b.The SAT Critical Reading scores for the senior class at Richmond Prep High is given in Table 3. Table 3: SAT Critical Reading Scores 520640750620470520 630600590660700580 460600640690530490 500560630760650760 580610710610590570 590550610490630550 590620610600570690

35. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) Solution a.In order to calculate the standard deviation using a TI-30XIIS/B calculator, begin by clearing the data lists in the calculator. Then enter the data points as before by using the following commands: 1. Press. 2. Choose 1-VAR and press. 3. Press ( X = should appear.

36. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) 4.Enter a data value and press the down arrow key twice, since the frequency is one for each data point. 5.After the last data point is entered, press. Because the values given are only a sample of students, we want the sample standard deviation. To calculate the sample standard deviation, press. Scroll over to the sample standard deviation, which is denoted by sx in the list of calculated values. From the list we see that s ≈ 10.3.

37. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) To calculate the standard deviation using the list function on a TI-83/84 Plus calculator, 1.Press, then choose 1:Edit..., and enter your data in L1. 2.Press again and now scroll to the right to CALC.

38. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) 3.Choose option 1:1-Var Stats and press. If your data are in L1, press again since L1 is the default list. If you did not type your data in L1, enter the list where your data are located, such as L2 or L3. (These list names are in blue, above the numeric keys.)

39. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) A list of numerical statistics will be generated for the data. The beginning of the list is shown in the margin. Because the values given are only a sample of college students, we want the sample standard deviation, which is denoted by s (on the calculator, this is displayed as Sx). From the list we see that s ≈ 10.3. Since the standard deviation tells us about the average distance away from the mean, we can conclude that student tweeting behavior usually varies from the mean by tens rather than hundreds of tweets.

40. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) b.Begin by clearing the data lists in the calculator. Now enter the data as you did in part a. Since we are told that the values given represent an entire Senior class, we want the population standard deviation, which is denoted by  x. From the list we see that  ≈ 72.8.

41. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 7: Calculating Standard Deviation Using a Calculator (cont.) Therefore, we know that SAT Critical Reading scores differ from the mean on average by 72.8 points. While we’ve got the calculator handy, we can see that the mean is actually approximately 602.9. Although there is not an actual score of 602.9, you can see that many of the students scores fall within about 70 points (or 1 standard deviation) of that mean, either larger or smaller.

42. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Find the population standard deviation for the following data. 8, 12, 10, 11, 13, 12, 15, 9, 11, 16 Answer: 2.4

43. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Percentile Percentiles divide the data into 100 equal parts and tell you approximately what percentage of the data lies at or below a given value.

44. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 9: Interpreting Percentiles Sierra received her scores from taking a mathematics placement test for her chosen university. Choose the best explanation for what it means for her to be in the 61st percentile. a.She correctly answered 61% of the answers on the test. b.61% of people taking the test scored the same as Sierra. c.Sierra’s score was at least as good as 61% of the people taking the test. d.Sierra missed 39% of the test questions.

45. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 9: Interpreting Percentiles (cont.) Solution The correct interpretation of her score is c.: “Sierra’s score was at least as good as 61% of the people taking the test.” Both a. and d. are incorrect because they refer to how many questions she answered correctly on the test and not how she did in comparison to others taking the test. b. is not quite correct because percentiles tell you the percentage that scored at or below you. They are not all necessarily the same score as Sierra’s.

46. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Quartiles Q 1 =First Quartile = 25 th percentile, that is, 25% of the data is less than or equal to this value. Q 2 =Second Quartile = 50 th percentile, that is, 50% of the data is less than or equal to this value. Q 3 =Third Quartile = 75 th percentile, that is, 75% of the data is less than or equal to this value. By definition, Q 2 will be the same as the median.

47. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 10: Interpreting Quartiles On Karl’s recent standardized test results, the picture graph of his score showed he was above the third quartile in language arts. His classmate, Asher, said his score was at the 70 th percentile, while Rylie said hers was at the 79 th percentile. Which of the three had the best language arts test score? Solution We know the percentile ranks of both Asher and Rylie are the 70 th and 79 th respectively. What we know about Karl’s score is that it was above the third quartile.

48. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 10: Interpreting Quartiles (cont.) Since the third quartile is the same as the 75 th percentile, we know that his score was somewhere at or above the 75 th percentile. We can conclude that he did better than Asher, whose score was at the 70 th percentile, but can make no definite comparison with Rylie, whose score was at the 79 th percentile, because we do not know for sure which one had the best language arts score.

49. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Box Plot Five Number Summary: Min, Q1, Q2, Q3, Max Q2 is same as median. Use my Statistics slides on Box Plot.